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Games, Puzzles, and Computation by Robert Aubrey Hearn B.A., Rice University (1987) S.M., Massachusetts Institute of Technology (2001) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2006 c Massachusetts Institute of Technology 2006. All rights reserved. Author.............................................................. Department of Electrical Engineering and Computer Science May 23, 2006 Certified by. Erik D. Demaine Esther and Harold E. Edgerton Professor of Electrical Engineering and Computer Science Thesis Supervisor Certified by. Gerald J. Sussman Matsushita Professor of Electrical Engineering Thesis Supervisor Accepted by......................................................... Arthur C. Smith Chairman, Department Committee on Graduate Students Games, Puzzles, and Computation by Robert Aubrey Hearn Submitted to the Department of Electrical Engineering and Computer Science on May 23, 2006, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science Abstract There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connec- tion is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games. In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis. In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Konane, Cross Purposes, TipOver, and others. Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman’s Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs. Thesis Supervisor: Erik D. Demaine Title: Esther and Harold E. Edgerton Professor of Electrical Engineering and Com- puter Science Thesis Supervisor: Gerald J. Sussman Title: Matsushita Professor of Electrical Engineering 2 Acknowledgments This work would not have been possible without the contributions of very many people. Most directly, I started on this line of research at the suggestion of Erik Demaine, who mentioned that the complexity of sliding-block puzzles was open, and that Gary Flake and Eric Baum’s paper on the complexity of Rush Hour might be a source of good ideas for attacking the problem. This intuition proved to be spot-on, but neither Erik nor I had any idea how many more results would follow in natural progression. But before this work began, I was already well-suited to head down this path. I acquired an early interest in games and puzzles, particularly with a mathematical fla- vor, primarily through Martin Gardner’s “Mathematical Games” column in Scientific American, and his many books. My early interest in the theory of computation is harder for me to pin down, but it was certainly dependent on having access to com- puters, which most children of my generation did not. I have my parents, particularly my father, to thank for this, as well as for exposure to the Martin Gardner books. During high school my good friend Warren Wood was a fellow traveler; we invented many an interesting (and often silly) mathematical game together. Later, I grew to love the beautiful mathematics of Combinatorial Game Theory, developed by Elwyn Berlekamp, John Conway, and Richard Guy. I am also grateful for many enjoyable and useful discussions with Elwyn, John, and Richard, which arose later during this work’s development. My interest in the theory of computation was rekindled when I returned to school after several years in the software industry, and took Michael Sipser’s Theory of Computation course. Before this, I viewed NP-completeness as deep, mysterious math. I understood the concept, but the thought that I might someday show a real problem to be NP-complete—or even harder—was not one I had seriously entertained. In Michael’s course I gained a clearer appreciation for how such reductions were done. It was in this context that I had the good fortune to TA MIT’s Introduction to Algorithms course for Charles Leiserson and Erik Demaine. I was put in charge of the class programming contest, and I chose puzzle design as the domain. This led to discussions of game and puzzle complexity with Erik, and eventually to all of the present work. I also learned the great value of teaching from Charles and Erik. There is nothing like having to teach MIT students algorithms to keep you on your toes and make the material come alive for you. Meanwhile, however, I was working on my “primary” research, implementing Mar- vin Minsky’s “Society of Mind”. I thank Marvin for his encouragement, and Gerry Sussman for his patience as I discovered as so many before me have that solving AI is not the task of a few years in grad school. I also have Gerry to thank for ultimately encouraging me to assemble my game and puzzle work into a Ph.D. thesis, and I also thank Erik for respectfully refraining from a similar suggestion until I raised the possibility with him, based on his understanding of my reason for being at MIT. After my initial results on puzzles, Albert Meyer and Shafi Goldwasser served as my RQE committee, and offered many useful perspectives, as well providing me with an extra sense of the legitimacy of my work and my ability to communicate it. 3 I am grateful to my coauthors on the work presented here: Erik Demaine, Michael Hoffman, Greg Frederickson, Martin Demaine, Rudolf Fleischer, and Timo von Oertzen. I learned a lot through collaboration that it would be virtually impossible to learn otherwise. Marty Demaine has also always had something interesting going on to discuss, and has offered excellent life advice on many occasions. As I began to get results, I met many other interesting people in the mathematical games community. I am especially fortunate to have met John Tromp and Michael Albert, who have become good friends, in addition to providing many valuable in- sights. Other “games people” I have enjoyed many discussions with and learned from include Richard Nowakowsi, Aviezri Fraenkel, J. P. Grossman, Aaron Seigel, Cyril Banderier, and Ed Pegg. I want to especially thank Ivars Peterson for helping to popularize some of my work in Science News and Math Trek, and making me aware of the wider interest in this kind of work. Similarly, thanks are due to Michael Kleber for inviting me to write an article for Mathematical Intelligencer. Of my many fellow grad students at MIT, I want to thank Jake Beal, Justin Werfel, Attila Kondacs, Ian Eslick, Radhika Nagpal, Paulina Varshavskaia, Rebecca Frankel, and Keith Winstein for helping to make the journey more pleasant. Two former students, Erik Rauch, who was my officemate, and Push Singh, who was a good friend and sometimes mentor in my Society of Mind work, both left this world before their time. Both, however, had a big positive effect on me, and many others. I want to thank Tom Knight, Norm Margolus, and Ed Fredkin for enlightening conversations about reversible computing, among other topics. My wife Liz deserves special thanks for being patient as I kept trying to solve impossible problems on the one hand, and screwing around with silly games on the other. Finally, Liz, I’ve made something of all that playing around with games! Last but not least, I am very appreciative of my committee. Gerry Sussman was my advisor from the moment I arrived at MIT, and it has been my great privilege to share many hours of discussion with Gerry on virtually every subject that is interest- ing, and many evenings of fine astronomy as well. Erik Demaine has been incredible to know and to work with. He has an amazing ability to point people in just the right direction, so that they will find interesting things. Every time I had a new result, I would take it to Erik, and he would say,“yes, cool! Now, did you think about this?”, and I would be off on another hunt. Marvin Minsky has been a major source of inspiration for me for more than 20 years. I hope to revive my Society of Mind work; I still feel that this book is the single best source of insights on how to think about intelligence. I am still somewhat astounded that I am able to just talk to Marvin like an ordinary person. Similarly, it has been an honor to have Patrick Winston on my committee. I have learned a lot about effective writing and communication, as well as AI, from Patrick. Marvin and Patrick deserve extra thanks for remaining on my committee when I switched topics from AI to game complexity. Finally, Michael Sipser, though a late addition to the committee, has been a valuable presence. I learned a great deal in his course—including a lot that I thought I already knew! And Michael also had valuable comments for me when he graciously agreed to join my committee, in spite of his load as head of the Mathematics department.

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